# Analyzing the earth conductivity and permittivity frequency dependence influence to electromagnetic transient phenomena

**ABSTRACT** In this article the quasi-modes model is used to observe the

influence, in electromagnetic transient phenomena, of considering a more

accurate representation of soil, taking into account the earth

conductivity and permittivity frequency dependence. For an actual 440 kV

three-phase transmission line the soil behavior is represented through a

unique real value of conductance and through a more accurate model,

considering its electromagnetic behavior

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**ABSTRACT:**The paper discusses the basic aspects of modeling, simulation and optimization o f transmission lines, with emphasis in validity, applicability and limitations of some used pro- cedures, and some guidelines for development of new procedures. 1. INTRODUCTION In this paper we discuss the basic aspects of modeling, simulation and optimization of transmission lines. As the subject can not be systematically covered within a single paper, we have chosen some specific points, as examples, which we discuss with emphasis in validity, applicability and limitations of some used procedures, and some guide- lines for development of new procedures. Due to the vast bibliography about lines and its modeling, following the aim of this paper, we do not give a comprehensive list of references. We only indicate, also as examples, some refer- ences directly related with chosen specific points and the way they are presented. By no means, the choice of topics may be interpreted as a judgment of methodologies and procedures not dealt with in the paper. The topics related to transmission lines modeling, simula- tion and optimization cover : a. Physical evaluation of basic line parameters and associ- ated analytical, numeric, analogous or test procedures, and accuracy requirements, according to foreseen applications. b. Procedures to represent a line, considering its basic parameters and length, e. g. as seen from its terminals, and in a form adequate for the specific application. c. Modeling and simulation of a line inserted in a network and interacting with the behavior of other elements. d. Optimization of a line, according normal and transient operational constraints and effects, investment and opera- tional cost, reliability, service quality, people and equip- ment safety, ambient impact, effects of eventual deviations from foreseen operational requirements. In what concerns operational aspects and constraints, the following aspects must be taken into account: A. In what concerns electromagnetic behavior: - SourceAvailable from: Maria Cristina Dias Tavares
##### Conference Paper: Influence of accurate soil representation for transmission-line parameters: Analyses based on Carson's modified formulations

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**ABSTRACT:**In this paper the influence of earth's conductivity and permittivity frequency dependence are evaluated when calculating transversal and longitudinal transmission lines' parameters directly from numerical integration of Carson's modified expressions. As an example of the importance of properly considering the frequency-dependent soil model, an actual 440 kV single three-phase transmission-line was represented, comparing the longitudinal and transversal parameters considering the earth's conductivity and permittivity frequency dependence soil model in relation to the common soil representation with a constant conductivity and permittivity that can be neglected assuming a low frequency approximation.Transmission and Distribution Conference and Exposition: Latin America (T&D-LA), 2010 IEEE/PES; 12/2010 - SourceAvailable from: Jerzy Wtorek

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

Influence of Earth Conductivity and Permittivity Frequency

Dependence in Electromagnetic Transient Phenomena

Carlos Medeiros Portela1, Maria Cristina Tavares2 and José Pissolato Filho2

(1) COPPE – Federal University of Rio de Janeiro, Rua Eng. Cesar Grillo, 249, Rio de Janeiro, RJ,

22640-150, BRAZIL e-mail : portelac@ism.com.br), (2) School of Electrical and Computer Engineering,

Dept.of Energy Control and Systems, State University of Campinas, PO Box 6106, 13081-970, Campinas,

SP, Brazil (e-mail:cristina@dsce.fee.unicamp.br, pisso@dsce.fee.unicamp.br),

The content of the paper is as follows : in Section 2

the soil electromagnetic behavior is described, with the

presentation of some measured results; in Section 3 line

parameters are calculated; in Section 4 an application to

an actual transmission line is presented.

Abstract – In this article a more accurate representation of

soil behavior is presented. The proposed model takes into

account the earth conductivity frequency dependence and

the earth permittivity, which normally are not considered.

One of the aspects covered in the paper is the importance

of properly considering the earth’s electromagnetic behav-

ior when calculating transmission line parameters. For an

actual 440 kV three-phase transmission line the soil behav-

ior is represented through a unique real value of conduc-

tance ( the normal approach ) and through the proposed

model.

II. SOIL ELECTROMAGNETIC BEHAVIOR

One essential aspect of grounding systems study and

simulation is adequate soil modeling.

Except for very high electric fields, that originate sig-

nificant soil ionization, soil electromagnetic behavior is

essentially linear, but with electric conductivity, σ , and

electric permittivity, ε , strongly frequency dependent.

The magnetic permeability, µ , is, in general, almost

equal to vacuum magnetic permeability, µ0. For slow

variation of electromagnetic entities, a hysteresis type

behavior may occur. For direct current or very slow

variations of electromagnetic entities, humidity migra-

tion phenomena, including electroosmosis and effects of

temperature heterogeneity may take place, which cannot

be dealt with only by means of local soil parameters.

For fast transients, namely those associated to light-

ning, the soil behavior is important in a reasonably wide

frequency range, typically from 0 to 2 MHz .

Keywords : Soil model, Line parameters, Frequency depend-

ence, Electromagnetic transients.

I. INTRODUCTION

One essential aspect of transmission line modeling is

the adequate representation of ground, which has a big

influence in line parameters, ahead of being a dominant

aspect for analysis and for design of line grounding sys-

tem. By historical and cultural reasons, most used pro-

cedures assume that the ground may be considered as

having a constant conductivity, frequency independent,

and an electric permittivity that can be neglected

( ω ε << σ ). These two assumptions are quite far from

reality, and can originate inadequate line modeling.

In the present paper a new soil model is presented.

This satisfies the physical coherence conditions concern-

ing the relation between conductivity (σ) and permittiv-

ity (ε) in the frequency domain. Some examples of

measured ground parameters are presented. The effect of

the soil behavior in some transmission line transients

and its influence on the overvoltages obtained are dis-

cussed.

For an actual 440 kV three-phase transmission line

the soil behavior is represented through a unique real

value of conductance, the most common assumption,

and through a more accurate model of its electromag-

netic behavior in relation to the earth conductivity and

permittivity frequency dependence.

In some cases, a proper earth model can lead to very

different results than the ones obtained with a simple

real conductivity value, as shown. The influence of the

frequency dependence of the soil parameters, in some

line transient phenomena, is analyzed.

A. Field Measurement Procedure

Field measurement procedures have been chosen after

measurement tests covering a large number of soil struc-

tures and conditions. The basic aspects related to col-

lecting the samples are due to the necessity of [1-5]:

- Assuring maintenance of natural soil consistence

and humidity, with sample material “identical”

to natural ground.

- Avoiding influence of small depth surface effects,

such as sun, wind and vegetation. These effects

may originate an important dispersion, in time

and space, and special measurement difficulties.

To consider such effects correctly, special meth-

ods, considering statistical distribution with

space and time correlation, may be required. In

most applications the error resulting of neglect-

ing such effects is relatively small.

- Avoiding important effects of local soil heteroge-

1

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

neity.

- Limit measurement errors related to electrode

shape and contact conditions between electrodes

and soil material.

Three basic procedures where adopted, namely :

- For rock, cylindrical samples (with 0.1 m diame-

ter with 0.8 m length) are obtained with a boring

machine.

- For reasonably consistent soils, a cutting and col-

lecting procedure is applied, obtaining samples

with a cuboid shape (1.2 m x 0.2 m x 0.2 m)

which are covered with a net, paraffin and a

wood box.

- For sand an pulvurulent soil, samples are col-

lected with a plastic tube with diameter 0.2 m

and 1.2 m long, to which steel pieces are

adapted to obtain easy penetration in soil and

sample cutting in tube extremity.

Two current copper plate electrodes (CE) are adapted

at sample extremities (with adjusted pressure) and two

copper cylindrical voltage electrodes (VE) are inserted,

with exemplificative geometry as in Figure 1. Through

an oscillator with variable frequency f , it is imposed the

current through the sample. From voltage at shunt ter-

minal and voltage between voltage electrodes (both

measured in amplitude and phase), and geometric fac-

tors, it is obtained

.

ωε+ σ

i

Figure 1- Schematic representation of a soil sample for meas-

urement of

in function of frequency.

ωε+σ

i

The field measurements of real soil have inherent dis-

persion. A purely mathematical fitting may lead to

physically inconsistent models with quite wrong results,

e.g. by Fourier methods. It is adequate to have a robust

validation criteria of soil models, covering real soil char-

acteristics.

In [1-5] several soil electric models have been pre-

sented and justified, which :

- Cover a large number of soil measured parame-

ters, with good accuracy, and within the range of

confidence of practical field measurement.

- Satisfy coherence conditions.

In this paper the electrical soil parameters are applied

(σ , ω ε ), in function of frequency, considering a par-

ticular set of the models described in [1-5]. The parame-

ters of such models were chosen according to a mini-

mum difference criterion for field measured electrical

parameters, in function of frequency, for 68 ground

samples at eight sites, in Brazil, covering very different

soil types and geological structures. The agreement of

obtained models with measured parameters is within or

near the confidence range of field measurement values.

The measurements were carried out in a frequency range

from 100 Hz to 2 MHz . At each site, the maximum dis-

tance between ground points at which samples were

collected was less than 500 m.

To show that the influence of small depth surface ef-

fects can be neglected in most applications, we indicate,

in Table 1, the soil depth d at which electromagnetic

field related to longitudinal line parameters reduces to

about 5 % of field at soil surface, in four examples, for

three frequencies, f . The first two (examples 1 and 2)

consider, respectively, soil 1 and soil 2 of item II E. of

this paper. The last two (examples 3 and 4) consider

soils similar to examples 1 and 2, but with a low fre-

quency conductivity of 1 mS/m.

Table 1 – Soil penetration depth, d , in four examples, for

three frequencies

d [m]

Example 1 Example 2 Example 3 Example 4

21 742 21 218

1 694 1 644

48 164

f = 60 Hz

f = 10 kHz

f = 1 MHz

6 169

510

6 155

477

40 48

B. Soil Models

The models which have been used in the presented re-

sults are some of the models described in [1-5].

With the exception indicated below, the models,

whose results are presented, are a sum of minimum

phase shift parcels, Wj , which apply to the immittance

type magnitude (in complex or tensorial formulation of

alternating magnitudes)

ωεσ

iW

+=

(ω = 2 π f , f being the frequency) ( 1)

where i = + −1 and

∑

=

j

=

m

j

W

0

W

( 2)

All submodels used for Wj are particular conditions of

a Type 3 model described in [1], presented below.

Apart from slow phenomena and hysteresis type phe-

nomena, soil behavior is, typically, of minimum phase

shift type. For a great number of soils, on frequency

range ( 0 , 2 MHz ), in a first approach, it is

α

ωσ⋅+=

ba

and

ωε= c

where a, b, c, α are constant parameters (frequency in-

dependent).

For some soils, a similar behavior occurs, but for a

smaller frequency range, e.g. ( 0 , 100 kHz ), and for

higher frequencies, the behavior is different, namely

with a lower ωε increase, or until a ωε decrease, when

frequency increases.

In order to analyze the frequency behavior of σ , ε , it

is convenient to consider complex formulations of elec-

tromagnetic entities, and to consider σ

mittance. In fact, apart from geometric factors,

may be associated with the admittance of a volume ele-

ment δv.

A type 3 model can be described as presented below,

for which :

ωα

( 3)

α

ω⋅

as an im-

+σ

ωε+ i

ωε

i

−

⋅=ω

ω

α+α

α

α+α

αα

a i

,,

b i

,,

ab

k)(Wj

, ,

1111

1212

FF

( 4)

2

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

representing 2F1[…,…,…,…] the hypergeometric func-

tion, with four arguments, 2F1 , according to the notation

of [6].

This submodel has four independent parameters (k, α,

a , b).

Considering, in this model (4), a = 0 , the model be-

comes :

⋅=

α

ω

+

ω

α

ω

(

α

b

1, 1F12

i ,

, )

b

kWj

=

, 1

+

ω

α

α

b

1F12

i ,

,

1 k

( 5)

Considering, in the model (4) , a = 0 , b → ∞ and

αj = α, the model becomes :

π

ω

2

A parcel Wj as indicated in (6) is equivalent to parcel

α

ω⋅

b

ωε⋅= c

⋅=

tanKj

α

ω

α

j

j

tan +1)(

⋅

⋅=

iKW

jj

( 6)

of σ and to , as indicated in (1) and

j

and αj = α, with the (3), with b = K

α

ω

π

j ,

c

α

2

condition

α

π

2

= tan

b

c

. This condition has been veri-

fied in soil measurements, within measurement accuracy

and soil heterogeneity effects.

Considering, in this model (6), αj = 0 , the model be-

comes :

σ constant, ω ε null (“pure” conductor) ( 7)

Considering, in the model (6), αj → 1 , the model be-

comes :

σ null, ω ε proportional to ω, ε constant (“pure” di-

electric) ( 8)

↑ σ σ [mS/m]

Log10 ( f / Hz )Log10 ( f / Hz )

↑ ωεωε [mS/m]

Figure 2 – Electric parameter of soil sample, f in logarithmic

scale.

Within the range ( 0 , 2 MHz ) , for all soil samples

modeled in this paper, it is accurate enough to consider

two parcels, for σ

, one constant (in most cases

real), and the other of type (4) or of type (5), frequency

dependent. In a few cases, there is a net hysteresis effect,

that can be modeled with an imaginary part of the con-

stant parcel. For all samples, α is the dominant parame-

ter of the relative shape of a frequency dependent parcel,

W

. For α = 0 such a parcel corresponds to

a “pure” conductor ( σ frequency independent, ε null ).

For α = 1 , such a parcel corresponds to a “pure” dielec-

ωε+ i

j, of σ

tric ( σ null , ε constant ). In all samples, for a frequency

dependent parcel, it is 0 < α < 1.

ωε+ i

n

α α

Physical soil structure without

major differences among the ten

soil samples at Site 4 .

Figure 3 – Electric parameter of soil sample, f in logarithmic

scale.

C. Soil Samples

In this section it is presented in graphic form σ σ and

ωεωε , in function of frequency, f , for models of one of

the ground samples obtained in Amazon region. In Fig-

ure 2 f is represented in logarithmic scale. In Figure 3 it

is represented, in bar form, the distribution of parameter

α of the function relating σ

(5) or (6).

to f , either of Type

ωε+ i

D. Statistical Distribution of Soil Parameters

In order to allow a direct interpretation of statistical

distribution of the main electric parameters of ground, in

a way which is independent of the model details, the

following parameters were chosen, according to the

models adopted, independently, for the 68 soil samples,

satisfying physical coherence conditions:

σ0 = σ (100 Hz) , σ at 100 Hz.

∆r = ∆σ1 = σ (1 MHz) - σ (100 Hz) σ,

increase between 100 Hz and 1 MHz.

∆i = ∆(ω ε)1 = ω ε (1 MHz) - ω ε (100 Hz),

ω ε increase between 100 Hz and 1 MHz.

α parameter of the frequency dependent parcel of

σ + i ω ε.

It was verified that, for these samples, the two parcels

of

ωε+σ

i

, one constant, the other frequency dependent,

are statistically independent. This fact, and the fact that

no significant correlation exists between the pair [∆i ,

α ] , although it exists between the pair [ ∆r , α ] , gives

rise to the hypothesis that:

- The constant and the frequency dependent parcels

of

ωε+σ

i

are related to quite distinct aspects

of physical ground behavior.

- The frequency dependent parcel is mainly associ-

ated with a dielectric physical process, with re-

lated dissipative effects. Such dissipative effects

are quite different from conductive behavior as-

sociated with the constant parcel.

In Figure 4 we represent the probability density, p , of

parameters σ0 , ∆r , ∆i , α , considered separately, and, in

Figure 5, the probability density, p , of parameters [∆i ,

α] , considered together, with Weibull approximations

based on the 68 soil samples.

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

p

σ σ0 [(mS/m) -1]

∆r [(mS/m)-1]

p

∆i [(mS/m) -1]

p

p

α α

Figure 4- Probability density, p , of parameters σ0 , ∆r , ∆i , α,

considered separately, with Weibull approximations based in

the 68 soil samples. Scales of p applicable to σ0 , ∆r , ∆i are

graduated in (mS/m)-1 .

α α

0.28

0.270.25

0.2

0.15

0.1

0.03

0.01

0.003

0.001

∆i [(mS/m)-1]

Figure 5- Probability density, p , of parameters [ ∆i , α ] ,

considered together, with Weibull approximations based in the

68 soil samples and without correlation between ∆i and α .

Values of p , in white, are expressed in (mS/m)-1 .

E. Soil Parameters Applied

The soil parameters used in these examples were ob-

tained from the experiments described in [3], and are

presented below :

- Soil 1 :

= Α + Β ω

ωε+σ

i

with A, B, α constants, and

A = 84.16 µS/m

B = [0.057849 + 0.12097 i] (µS/m) sα

α = 0.71603

- Soil 2 : σ = Α ; ω ε = 0

which results in ρ : 11882 Ω.m, constant.

The conductivity of the studied soils were chosen to

be equal at low frequency, in order to compare the ob-

tained results with the results of traditional procedure

α

that assumes constant conductivity (as measured at low

frequency) and ω ε = 0 . Soil 1 considers two parcels Wj

as described in II.3, namely one constant parcel, A , of

the type (7) and a frequency dependent parcel, Β ωα, of

the type (6). In soil 2, only the first parcel, A, is consid-

ered. Although this constant parcel has an apparently

low value, it is not uncommon to measure condutivities

of this magnitude or lower in several Brazilian regions.

By example, for a specific upcoming 1000 km-long

transmission line, in the center of Brazil: in 20 % of the

line length conductivity was in the range or lower than

the adopted value; the minimum measured value was

near 43 µS/m.

III. CALCULATING THE LINE PARAMETERS

In order to implement the soil model, the line parame-

ters were calculated using the approximated formula

which includes the earth effect in longitudinal imped-

ance as the equivalent to having an ideal ground surface

at a depth D (complex) below physical ground surface

[7].

'

Dkm

d km

h k

h'k

h m

h'm

k

m

real ground

ideal ground

()

0

1

D

µωεω+σ

=

ii

'

h'k = hk + D'

Figure 6 - Conductors k and m position supposing the earth at

a complex depth D’.

The transmission line longitudinal impedance matrix,

per unit length, may be obtained considering:

Z0 = | Z0km | k, m = 1, 2, ..., n

where :

Z0km – longitudinal impedance matrix element, per unit

length;

n – total number of conductors

and

Z0km = Zintkm + Zextkm

where

Zintkm – internal impedance, per unit length, of conduc-

tor k, for k = m , 0 for k

Zextkm – external impedance, per unit length, between

( 9)

( 10)

m ;

≠

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

conductors k , m;

and

ωµ

=

km

km

ext

d

D

Zln

2

i

0

π

k, m = 1, 2, ..., n ( 11)

where Dkm and dkm are defined schematically in Figure 6,

and given by :

1

'

µωεωσ +

For the self terms (k = m)

hD'2

=

( 13)

dkm = rk (external radius of conductor k ) ( 14)

and

Zint = Rint + i Xint ( 15)

where

Rint – internal conductor resistance

Xint – internal conductor reactance

In Figure 7 and Figure 8 the per unit longitudinal pa-

rameters for the transposed line using both soil models

are presented.

()

i

0

i

=

D

( 12)

kkm

10

1

10

2

10

Frequency [Hz]

3

10

4

10

5

10

6

0,01

0,1

1

10

100

1000

Single three-phase transposed line

soil 1 - A - 84.16 µS/m

B - [0.057849 + 0.12097 i] (µS/m) s

α - 0.71603

soil 2 - σ = 84.16 µS/m; ε = 0

α

α = β - soil 1

zero - soil 1

α = β - soil 2

zero - soil 2

Resistance [ohm/km]

Figure 7 – Resistance per unit length comparing both soil

models.

10

1

10

2

10

Frequency [Hz]

3

10

4

10

5

10

6

1

10

Single three-phase transposed line

soil 1 - A - 84.16 µS/m

B - [0.057849 + 0.12097 i] (µS/m) s

α - 0.71603

soil 2 - σ = 84.16 µS/m; ε = 0

α

α = β - soil 1

zero - soil 1

α = β - soil 2

zero - soil 2

Inductance [mH/km]

Figure 8 – Inductance per unit length comparing both soil

models.

The difference between line parameters for the two

soil models is important, namely for the homopolar

mode (e.g., 37 % in the longitudinal resistance per unit

length, at 10 kHz). For fast transients, for which impor-

tant frequency range may include frequencies above

10 kHz, the difference between the two soil models may

also be important for non-homopolar modes. E. g. , for

100 kHz, there is an order of magnitude difference in

resistance per unit length, between the two soil models,

for non-homopolar modes. Typical cases in which fre-

quency range above 10 kHz is important are : transients

originating from lightning; front of wave aspects of tran-

sients associated with short-circuits. Such cases may be

quite important in what concerns insulation coordination.

IV. SINGLE THREE-PHASE LINE

APPLICATION

In Figure 9 the data of the three-phase line used to il-

lustrate the model are presented.

0.4 m

0.4 m

(7.51, 36.00)

(9.27, 24.07)

3.60 m

phase conductors : grosbeak

ground wires : EHS - 3/8"

line length : 400 km

sag at midspan

phase cond. : 13.43 m

ground wires : 6.40 m

Figure 9 - Schematic representation of the 440 kV three-phase

line.

The line parameters were calculated in the range of

10 Hz to 10 kHz. As it is a single line, to represent its

modes (exact ones for a transposed line and quasi-modes

for a non-transposed line) Clarke’s transformation ma-

trix was applied as explained in [8-10]. With the longi-

tudinal impedance and transversal admittance in mode

domain, the synthetic circuits were calculated, com-

posed of one cascade of π-circuits for each mode, each

representing 10 km length. The 10 km length π-circuit

represents properly the line response up to 7 kHz. The

line was supposed ideally transposed.

A. Frequency Analysis

A frequency scan analysis was performed for both

models where the sending terminal had a 1 V source and

the receiving end was open. The relations between the

line ends were analyzed in the range of 10 Hz to 7 kHz.

In Figure 10 the zero sequence response is presented for

the transposed line.

10100 100010000

0

1

2

3

4

5

6

Frequency scan

Transposed line

soil 2 (mod)

soil 1 (mod)

Reception voltage (module) [V]

Frequency [Hz]

Figure 10 - Zero sequence - Transposed line.

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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA

6

The results for both soil models are discussed below :

- The positive sequence response was similar for

both models.

- The zero sequence response for the frequency de-

pendant soil model is more damped than the one

which uses only constant conductance.

- The difference between the zero sequence re-

sponse for the two soil models is important, e.g.

14 % at 1000 Hz.

V. CONCLUSIONS

In some existing systems there are unexplained differ-

ences, sometimes of the order of magnitude, between

calculated and measured values, such as between in-

duced voltages, electric field in ground and transferred

voltages. These differences can be related to the soil

models used.

So, it is essential for most applications concerning

grounding systems, or involving electromagnetic phe-

nomena affected by ground, to adequately model the

ground behavior, including several aspects not consid-

ered in common practice.

For transmission lines, according to specific condi-

tions, and the phenomena being studied, it may be quite

important to correctly model the soil, considering fre-

quency dependence of

ωε+σ

i

We have presented some illustrative results for a

440 kV three-phase transmission line. The soil behavior

is represented through two alternative soil models. In the

first soil model we have considered an accurate soil rep-

resentation, satisfying coherence conditions and with

frequency dependent. In the second soil model,

we have considered a constant, frequency independent,

conductance and ω ε much lower than σ. The two mod-

els have similar behavior at low frequency, but quite a

distinct one at higher frequency.

In some cases, an adequate earth model can lead to re-

sults quite different from those obtained with the usual

procedure of considering the parameter σ of soil fre-

quency independent and parameter ε frequency inde-

pendent with a relatively small value. The conditions in

which such a difference can be important include the

following examples :

- Switching conditions in which an important homopo-

lar component may occur, either due to the spread of

switching of the three poles, or to fault conditions, and

in which the important frequency spectrum is not re-

stricted to extremely low frequencies ( < 1 kHz), and

includes frequencies up to about 10 kHz.

- Network sustained operation, faults and maneuvers in

which conditions occur near resonance, for the ho-

mopolar component, for frequencies not restricted to

extremely low frequencies ( < 1 kHz), and, e. g. , for

frequency between 1 and 10 kHz.

- Fast transients, for which important frequency range

may include frequencies above 10 kHz. In this case,

the difference between an accurate soil model and

usual assumptions may be important also for non-

homopolar modes. Typical cases in which frequency

.

ωε

i

+σ

range much above 10 kHz (for some conditions above

1 MHz) is important are: transients originated by

lightning; front of wave aspects of transients associ-

ated with short-circuits, transients is gas-insulated

substations. From the presented results it is expected

that, for transient phenomena with dominant fre-

quency spectrum till above 10 kHz, the distinct ho-

mopolar mode response may have quite important ef-

fects, not restricted to higher attenuation. By example,

overvoltage shape, namely in front of wave, may be

quite different, with important consequences in insula-

tion coordination.

ACKNOWLEDGMENTS

The measurement and modeling of soil samples of which

partial results have been used in this paper, were done under a

Contract of CISCEA, Comissão de Implantação do Sistema de

Controle do Espaço Aéreo (Air Space Control System Estab-

lishment Commission) with Fundação COPPETEC. The au-

thors thank CISCEA the permission to use such results.

The authors also thank the financial support received from

FAPESP - Fundação de Amparo à Pesquisa do Estado de São

Paulo and from CNPq – Conselho Nacional de Desenvolvi-

mento Científico e Tecnológico.

REFERENCES

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